One important method of analysing multiple thin layers on a substrate involves exposing a specimen to a beam of electrons and measuring the emitted x-ray spectrum. Provided the electron beam is sufficiently energetic to penetrate through the layers and reach the substrate, then characteristic x-rays are generated from elements in both the substrate and the various layers and these contribute to the total x-ray spectrum seen by an x-ray detector. FIG. 1 shows a typical situation where a 10 keV electron beam is incident on a layered sample with three layers of different thickness on a substrate. Many electron trajectories are shown and x-rays may be generated at any point along the electron trajectory as a result of ionisation of atoms. X-rays are emitted in all directions and if an x-ray detector is positioned above the sample, then x-rays emitted towards the detector will provide signals representative of the elements present in the regions excited by the electron beam. In the arrangement of FIG. 1, x-rays will emerge from all three layers and the substrate. In FIG. 2, the same sample is exposed to a lower energy electron beam. In this case, the electrons only penetrate to the top two layers so there will be no signal from the lowest layer and the substrate. In general, if a series of x-ray spectra are acquired at different incident electron beam energies, typically between 2 keV and 20 keV, then the corresponding spectra will exhibit characteristic x-ray peaks that vary according to the thicknesses and the compositions of the various layers, and indeed the substrate. Given a particular electron beam energy, the characteristic intensity for a particular element within a multi-layered sample can be expressed as a “k-ratio”.
A “k-ratio” is the ratio of the x-ray intensity received for a particular element from the structure (counts per second recorded from a characteristic x-ray emission series such as K, L or M) to that obtained from a flat bulk specimen of pure element under the same experimental conditions. Taking this ratio avoids having to know the x-ray detector collection efficiency as a function of energy. By measuring a series of k-ratios, it is sometimes possible to deduce the thicknesses and compositions of the various layers in a multi-layer specimen.
If NE elements in total occur in one or more of the layers or in the substrate and there are NL layers with thickness T1, T2 . . . TNL, and layer L contains concentration CLj of element j, and the substrate contains concentration CSj of element j then the predicted k-ratio for element i at incident electron beam energy E0 can be written as:—ki=fi(E0, T1, T2, . . . TNL, C11, C12, . . . C1NE, C21, C22 . . . C2 NE . . . CNL 1, CNL 2, . . . CNL NE, CS1, CS2, . . . CS NE)  EQUATION 1
where fi is a non-linear function of the layer thicknesses and the compositions of the layers and substrate. Several equations of this form cover the measured element intensities at this one beam energy E0. There may be more than one equation for a particular element if measurements are made on more than one emission series for this element (e.g. K or L or M emissions). Measurements may also be taken at further values of the beam energy and in general there will be M of these non-linear equations for k-ratios. The function in equation 1 typically involves integrations and non-linear functions and in general it is not possible to “invert” the set of equations and write down a formula that expresses the thickness or compositions for any one layer in terms of a set of measured k-ratios. Therefore, to determine a set of thicknesses and compositions (“layer variables”) from a set of x-ray measurements, it is known to use a modelling approach where the parameters of the model are adjusted to find a set of ki that are a “best-fit” to the measured k-ratios (see for example, Chapter 15 in “Numerical Recipes in C”, Second Edition, W.H. Press et al, Cambridge University Press 1999).
Thus, a computer program is used to make iterative guesses at the thickness and composition of the layers to find a set that is a close fit with the k-ratios measured from x-ray spectra (see for example, J. L. Pouchou. “X-ray microanalysis of stratified specimens”, Analytica Chimica Acta, 283 (1993), 81-97. This procedure has been made available commercially in the software product “Stratagem” by SAMx, France). The computer program will make a test at each iteration to see if the guesses are not changing significantly between iterations, in which case “convergence” is achieved.
Unfortunately in some cases, it is impossible to find a best-fit set of thicknesses and compositions because the measured k-ratios do not reveal enough differences in intensity to resolve the source of the individual contributions to x-ray intensity. In this case, the computer program iterations will fail to converge on a unique solution. Such problems can sometimes be resolved by choosing different x-ray emission series, different beam energies or constraining the range of possible solutions by providing information on some of the thicknesses or compositions where this is known beforehand. Although there are some guidelines for the choice of beam energies and x-ray series (see for example, J. L. Pouchou, “X-ray Microanalysis of Thin Films and Coatings”, Microchim. Acta 138, 133-152 (2002)), in general it is difficult to prove that a given type of sample can always be analysed successfully by this technique except by extensive experimentation in the hands of an expert. Existing software programs (such as “Stratagem”) provide diagnostic tools such as curve plots of k-ratio vs layer thickness or k-ratio vs beam energy in order to assist the expert in making a choice of the best conditions for analysis. However, these tools do not provide any recommendations for analysis, particularly for a complex multi-layered specimen, and are not suitable for a person that does not have a good understanding of the physics involved.
The iterative procedure of the computer program begins with a set of starting values for the fitted parameters (that is the “unknowns”) and the choice of starting “guesses” can affect whether the iteration will converge to the best solution for a given set of input data. Therefore, even when suitable conditions can be defined for the measurements and some prior knowledge is available, the procedure can still fail if the starting guesses for the fitted parameters are not suitable.
When an energy dispersive spectrometer is used to acquire an x-ray spectrum, the digitised spectrum effectively consists of a number of independent channels that each record counts for a small range of energies. To obtain peak intensities, the spectrum is processed mathematically to remove background and correct for any peak overlap. Each channel count in the original spectrum is subject to Poisson counting statistics and this will cause variation in the derived peak areas.
The physical theory for x-ray generation in layered specimens shows that a change in any one composition or thickness value will in general affect several of the observed x-ray intensities. Conversely, a change in one measured elemental intensity can influence several of the fitted parameters. In some situations, it is possible that some random perturbation of measured intensities together with the particular set of starting guesses for the fitted parameters will cause the program to sometimes succeed and sometimes fail to converge to a solution. In a quality control application where repeat measurements are taken to ensure thicknesses and/or compositions are within agreed tolerances, it is critical to ensure such failures are rare.
Even if convergence is always achieved, because of statistical variation each new set of measurements on the same specimen will give rise to a different set of results for thicknesses and composition. Using longer x-ray measurement times will in general reduce the fluctuation in measured k-ratios and this will in turn reduce the variation in thickness and composition estimates obtained by iteration. However, because of the complicated and mathematically non-linear nature of the problem, it is not possible to formulate equations to predict what precision can be achieved in the measured thicknesses or compositions.
A standard method to obtain estimates of reproducibility for such non-linear problems is to use Monte Carlo simulation (see “Numerical Recipes in C”, Second Edition, W.H. Press et al, Cambridge University Press 1999, p 689). In this method, a set of results is first obtained from a set of input data. A set of random numbers representing the appropriate statistical distribution is added as a perturbation to the input data and the results found by iterative solution. These results will be slightly different from the results without the perturbations and the differences are recorded. This process is repeated many times with new sets of random numbers and the changing output results can be used to calculate the standard deviation for each result value as a consequence of counting statistics affecting the input data. For example, this sensitivity analysis approach has been used in the specialised field of x-ray fluorescence analysis where x-rays from an x-ray tube are used to excite an x-ray spectrum from the specimen (U.S. Pat. No. 6,118,844). In U.S. Pat. No. 6,118,844 individual channel counts in the uncorrected x-ray data are subjected to counting statistic perturbations to predict the effect, after corrections for peak overlap, on the uncertainty in calculated layer thickness by the fluorescence method.
In the practical situation of a materials analyst being presented with a new type of thin film sample to be studied, experimental spectra usually have to be acquired from a similar but known sample to find the observed x-ray intensities from the various elements. Further experiments are usually then required to decide on the best set of measurement conditions, the element series to be measured, the fixed and variable parameters and the starting values. Thus it will be appreciated that the procedure to establish the feasibility of the electron beam analysis approach can be extremely time consuming and expensive. Furthermore, it requires considerable knowledge on the part of the operator There is therefore a desire to improve upon these known methods of determining the feasibility of performing electron beam x-ray analysis.